A036065 Triangle of flag coefficients [ n k ] (denominators of rational parts).

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 3, 1, 1, 3, 1, 1, 16, 1, 8, 1, 16, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 32, 1, 32, 3, 32, 1, 32, 1, 1, 35, 1, 5, 1, 1, 5, 1, 35, 1, 1, 256, 1, 64, 1, 128, 1, 64, 1, 256, 1, 1, 63, 1, 7, 3, 1, 1, 3, 7, 1, 63, 1, 1, 512, 1, 512, 1, 256, 5, 256, 1, 512, 1, 512

Offset

0,5

References

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 69.

Links

Table of n, a(n) for n=0..89.

Formula

[ 2m 2k ] = C(2m, 2k)/C(m, k); [ 2m 2k+1 ] = Pi(2m)!/4^m*k!m!(m-k-1)!; [ 2m+1 2k ] = 4^k*C(m, k)/C(2k, k); [ 2m+1 2k+1 ] = 4^(m-k)*C(m, m-k)/C(2m-2k, m-k).

Example

Triangle begins
  1;
  1 1;
  1*Pi/2 1;
  1 2 2 1;
  1 3*Pi/4 3 3*Pi/4 1;
  1 8/3 4 4 8/3 1; ...

Mathematica

f[m_?EvenQ, k_?EvenQ] := Binomial[m, k]/Binomial[m/2, k/2]; f[m_?EvenQ, k_?OddQ] := (Pi*m!)/(2^m*(((1/2)*(k - 1))!*(-1 + (1 - k)/2 + m/2)!*(m/2)!)); f[m_?OddQ, k_?EvenQ] := (2^k*Binomial[(1/2)*(m - 1), k/2])/ Binomial[k, k/2]; f[m_?OddQ, k_?OddQ] := (4^((1 - k)/2 + (1/2)*(m - 1))* Binomial[(1/2)*(m - 1), (1 - k)/2 + (1/2)*(m - 1)]) / Binomial[m - k , (1 - k)/2 + (1/2)*(m - 1)]; A036065 = Denominator[ Flatten[ Table[f[m, k], {m, 0, 12}, {k, 0, m}] ]](* Jean-François Alcover, May 10 2012, from formula *)

Crossrefs

Cf. A036064.

Sequence in context: A345932 A205399 A135303 * A082907 A146532 A305720

Adjacent sequences: A036062 A036063 A036064 * A036066 A036067 A036068

Keyword

nonn,easy,tabl,nice,frac

Author

N. J. A. Sloane

Extensions

More terms from Naohiro Nomoto, Jun 20 2001

Status

approved